How to not display surplus texture (UV mapping)
You have the correct node setup for the material, but the image you are using needs to be alpha masked.

That is to say, you would need to take the image into (for example) Photoshop, and "cut out" the eyeball so that it sits on a transparent background. You would then need to save this a file format that allows transparency (e.g. .PNG).

This isn't working because, at the moment, your eyeball sits on a black background. Your nodes tell Blender only to show the image when it has alpha (i.e. only show it where it is not transparent). This is correct, but the problem is that your image is not transparent anywhere, so all of it shows.